Left side rear turn signal and back-up light not working?

I have a 2004 SR5 4Runner. Have replaced turn signal bulb and socket for it. Tail and brake lights work fine. When I use L turn signal, the corresponding dash light flashes really fast. All fuses in both boxes (cab and under hood) look fine.
Hazards lights turned on ... the left rear light doesn't come on.
I also checked the turn signal bulb by placing it in right rear light socket - worked fine over on that side.
I haven't replaced the back up bulb. If that is bad, could that cause the turn signal bulb t not light up?
Thank you in advance.

 

Rapid flashing of turn signal typically associated with a burned out bulb.

 

Take a look at the front bulbs also for that side.

 

Front L turn signal build works fine. The new bulb I put in the rear L blinker socket is good, just won't come on in that socket. Thank you.

 

You said you replaced the socket for the L/R turn signal bulb and it's flashing fast. Are you able to take a voltage reading on the wire for that bulb? I think you have a broken wire some place in the harness. The backup and turn signal lights are independent of each other but since both are not working I believe it's wiring or the socket/connector to the taillight assembly. I've never opened mine up but I think there is a plug that goes onto the light assembly. That's where I would take voltage readings.